Typical faces of best approximating three-polytopes (Q2381810)

For \(K\) a convex body in \({\mathbb R}^3\) with \(C^2\) boundary, let \(P^c_{(n)}\) be a polytope of minimal volume with at most \(n\) facets circumscribed to \(K\). \textit{P. M. Gruber} [Geom. Dedicata 84, No.~1--3, 271--320 (2001; Zbl 0982.52020)] proved that the typical facets of \(P_{(n)}^c\) are asymptotically as \(n\to\infty\) close to regular hexagons in a suitable sense if the Gaussian curvature is positive on \(\partial K\). In the paper under review, the authors show that typical facets of inscribed polytopes with \(n\) vertices and maximal volume are asymptotically close to regular triangles in a suitable sense. They also prove analogous statements for general polytopes (not necessarily inscribed or circumscribed to \(K\)) with \(n\) vertices and with \(n\) facets that minimize the symmetric difference metric to \(K\).